Respuesta :
Answer:
a) 36.32% probability that a randomly selected pregnancy lasts less than 201 days.
b) 3.22% probability that a random sample of 16 preganacies has a mean geatation period of 201 days or less.
c) 1.58% probability that a random sample of 37 preganacies has a mean geatation period of 201 days or less
Step-by-step explanation:
To solve this problem, we have to know the normal probability distribution and the central limit theorem.
Normal probability distribution
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
Central Limit theorem
The Central Limit Theorem estabilishes that, for a random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], a large sample size can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex]
In this problem, we have that:
[tex]\mu = 207, \sigma = 17[/tex]
(a) what is the probability that a randomly selected pregnancy lasts less than 201 days?
Pvalue of Z when X = 201.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{201 - 207}{17}[/tex]
[tex]Z = -0.35[/tex]
[tex]Z = -0.35[/tex] has a pvalue of 0.3632.
So 36.32% probability that a randomly selected pregnancy lasts less than 201 days.
(b) what is the probability that a random sample of 16 preganacies has a mean geatation period of 201 days or less?
Now [tex]n = 16, s = \frac{17}{\sqrt{16}} = 3.25[/tex]
Again the pvalue of Z when X = 201
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
Due to the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{201 - 207}{3.25}[/tex]
[tex]Z = -1.85[/tex]
[tex]Z = -1.85[/tex] has a pvalue of 0.0322.
So 3.22% probability that a random sample of 16 preganacies has a mean geatation period of 201 days or less.
(c) what is the probability that a random sample of 37 preganacies has a mean geatation period of 201 days or less?
Now [tex]n = 16, s = \frac{17}{\sqrt{37}} = 2.79[/tex]
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{201 - 207}{2.79}[/tex]
[tex]Z = -2.15[/tex]
[tex]Z = -2.15[/tex] has a pvalue of 0.0158.
1.58% probability that a random sample of 37 preganacies has a mean geatation period of 201 days or less
